C₃²×#7099 #72245482

order	2304
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₁₆ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#7099

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	#106
commutator	C₈	C₂×C₃²×C₁₆
Frattini, cosocle	C₈²	C₂²×C₃²
socle, prime radical	C₂²×C₃²	#334
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ C₃²×C₈² ◁ G

294 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 2304

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,22,23,28)(19,25)(20,27)(24,31,30,29)(26,32)(33,34,35);(17,18,19,22,20,24,25,29,21,23,26,28,27,30,32,31)(36,37,38)

Conjugacy classes & character table

576 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₃⋊D₃×#7099, #72254841, #72269777, …
maximal subgroup of	C₃²×#93877, C₃⋊D₃×#7099, #72254841, …
maximal normal subgroup of	C₃²×#93877, C₃⋊D₃×#7099, #72254841, …
maximal quotient of	C₃²×#93877

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,22,23,28)(19,25)(20,27)(24,31,30,29)(26,32)(33,34,35), (17,18,19,22,20,24,25,29,21,23,26,28,27,30,32,31)(36,37,38));